Open Access
June 2010 Trek separation for Gaussian graphical models
Seth Sullivant, Kelli Talaska, Jan Draisma
Ann. Statist. 38(3): 1665-1685 (June 2010). DOI: 10.1214/09-AOS760


Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar d-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.


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Seth Sullivant. Kelli Talaska. Jan Draisma. "Trek separation for Gaussian graphical models." Ann. Statist. 38 (3) 1665 - 1685, June 2010.


Published: June 2010
First available in Project Euclid: 24 March 2010

zbMATH: 1189.62091
MathSciNet: MR2662356
Digital Object Identifier: 10.1214/09-AOS760

Primary: 62H99 , 62J05
Secondary: 05A15

Keywords: Bayesian network , Conditional independence , Gessel–Viennot–Lindström lemma , Graphical model , Linear regression , trek rule

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 3 • June 2010
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